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Relation between CS/WZW and AdS/CFT

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One precise example of realization of the holographic principle is the CS/WZW correspondence, which relates 3d Chern-Simons theory with the 2d Wess-Zumino-Witten model. As explained for example in this article in nlab there is a relation between this correspondence and the AdS3/CFT2 correspondence (which in turn relates 3d gravity and 2d conformal field theory). In particular CS/WZW appears to be some part (nlab says "sector") of the full AdS/CFT correspondence.

How does CS/WZW appear inside AdS3/CFT2? I'm asking, if possible, for a rough explanation of how this works. It seems that this happens in other dimensions too. Is there any similar TFT/CFT correspondence that can be seen as a sector of the usual AdS/CFT setting with $\rm{AdS}_5\times S^5$ type II supergravity / 4d $\mathcal{N}=4$ super Yang-Mills?

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Holographic duality doesn't necessarily require $AdS$ space. See, for example, the famous CS/Topological String duality. In particular, I don't think that it is possible to consider CS/WZW as $AdS_3/CFT_2$.

I'm aware that there are holographic dualities that don't involve AdS. CS/WZW is one example. However, it seems that AdS3/CFT2 has some sector that can be identified with CS/WZW (see the link in my question). Notice that I'm not saying that those two dualities are the same, but that there is some relatiom between them.

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